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Nicolaou ZG, Bramburger JJ. Complex localization mechanisms in networks of coupled oscillators: Two case studies. CHAOS (WOODBURY, N.Y.) 2024; 34:013131. [PMID: 38252783 DOI: 10.1063/5.0174550] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/31/2023] [Accepted: 12/14/2023] [Indexed: 01/24/2024]
Abstract
Localized phenomena abound in nature and throughout the physical sciences. Some universal mechanisms for localization have been characterized, such as in the snaking bifurcations of localized steady states in pattern-forming partial differential equations. While much of this understanding has been targeted at steady states, recent studies have noted complex dynamical localization phenomena in systems of coupled oscillators. These localized states can come in the form of symmetry-breaking chimera patterns that exhibit coexistence of coherence and incoherence in symmetric networks of coupled oscillators and gap solitons emerging in the bandgap of parametrically driven networks of oscillators. Here, we report detailed numerical continuations of localized time-periodic states in systems of coupled oscillators, while also documenting the numerous bifurcations they give way to. We find novel routes to localization involving bifurcations of heteroclinic cycles in networks of Janus oscillators and strange bifurcation diagrams resembling chaotic tangles in a parametrically driven array of coupled pendula. We highlight the important role of discrete symmetries and the symmetric branch points that emerge in symmetric models.
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Affiliation(s)
- Zachary G Nicolaou
- Department of Applied Mathematics, University of Washington, Seattle, Washington 98195-3925, USA
| | - Jason J Bramburger
- Department of Mathematics and Statistics, Concordia University, Montréal, Quebec H3G 1M8, Canada
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2
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Lee S, Krischer K. Heteroclinic switching between chimeras in a ring of six oscillator populations. CHAOS (WOODBURY, N.Y.) 2023; 33:2894497. [PMID: 37276574 DOI: 10.1063/5.0147228] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/20/2023] [Accepted: 05/15/2023] [Indexed: 06/07/2023]
Abstract
In a network of coupled oscillators, a symmetry-broken dynamical state characterized by the coexistence of coherent and incoherent parts can spontaneously form. It is known as a chimera state. We study chimera states in a network consisting of six populations of identical Kuramoto-Sakaguchi phase oscillators. The populations are arranged in a ring, and oscillators belonging to one population are uniformly coupled to all oscillators within the same population and to those in the two neighboring populations. This topology supports the existence of different configurations of coherent and incoherent populations along the ring, but all of them are linearly unstable in most of the parameter space. Yet, chimera dynamics is observed from random initial conditions in a wide parameter range, characterized by one incoherent and five synchronized populations. These observable states are connected to the formation of a heteroclinic cycle between symmetric variants of saddle chimeras, which gives rise to a switching dynamics. We analyze the dynamical and spectral properties of the chimeras in the thermodynamic limit using the Ott-Antonsen ansatz and in finite-sized systems employing Watanabe-Strogatz reduction. For a heterogeneous frequency distribution, a small heterogeneity renders a heteroclinic switching dynamics asymptotically attracting. However, for a large heterogeneity, the heteroclinic orbit does not survive; instead, it is replaced by a variety of attracting chimera states.
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Affiliation(s)
- Seungjae Lee
- Physik-Department, Technische Universität München, James-Franck-Straße 1, 85748 Garching, Germany
| | - Katharina Krischer
- Physik-Department, Technische Universität München, James-Franck-Straße 1, 85748 Garching, Germany
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3
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Lee S, Krischer K. Chaotic chimera attractors in a triangular network of identical oscillators. Phys Rev E 2023; 107:054205. [PMID: 37328989 DOI: 10.1103/physreve.107.054205] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/22/2022] [Accepted: 04/17/2023] [Indexed: 06/18/2023]
Abstract
A prominent type of collective dynamics in networks of coupled oscillators is the coexistence of coherently and incoherently oscillating domains known as chimera states. Chimera states exhibit various macroscopic dynamics with different motions of the Kuramoto order parameter. Stationary, periodic and quasiperiodic chimeras are known to occur in two-population networks of identical phase oscillators. In a three-population network of identical Kuramoto-Sakaguchi phase oscillators, stationary and periodic symmetric chimeras were previously studied on a reduced manifold in which two populations behaved identically [Phys. Rev. E 82, 016216 (2010)1539-375510.1103/PhysRevE.82.016216]. In this paper, we study the full phase space dynamics of such three-population networks. We demonstrate the existence of macroscopic chaotic chimera attractors that exhibit aperiodic antiphase dynamics of the order parameters. We observe these chaotic chimera states in both finite-sized systems and the thermodynamic limit outside the Ott-Antonsen manifold. The chaotic chimera states coexist with a stable chimera solution on the Ott-Antonsen manifold that displays periodic antiphase oscillation of the two incoherent populations and with a symmetric stationary chimera solution, resulting in tristability of chimera states. Of these three coexisting chimera states, only the symmetric stationary chimera solution exists in the symmetry-reduced manifold.
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Affiliation(s)
- Seungjae Lee
- Physik-Department, Technische Universität München, James-Franck-Straße 1, 85748 Garching, Germany
| | - Katharina Krischer
- Physik-Department, Technische Universität München, James-Franck-Straße 1, 85748 Garching, Germany
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Cabral J, Castaldo F, Vohryzek J, Litvak V, Bick C, Lambiotte R, Friston K, Kringelbach ML, Deco G. Metastable oscillatory modes emerge from synchronization in the brain spacetime connectome. COMMUNICATIONS PHYSICS 2022; 5:184. [PMID: 38288392 PMCID: PMC7615562 DOI: 10.1038/s42005-022-00950-y] [Citation(s) in RCA: 19] [Impact Index Per Article: 9.5] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 11/19/2021] [Accepted: 06/20/2022] [Indexed: 01/31/2024]
Abstract
A rich repertoire of oscillatory signals is detected from human brains with electro- and magnetoencephalography (EEG/MEG). However, the principles underwriting coherent oscillations and their link with neural activity remain under debate. Here, we revisit the mechanistic hypothesis that transient brain rhythms are a signature of metastable synchronization, occurring at reduced collective frequencies due to delays between brain areas. We consider a system of damped oscillators in the presence of background noise - approximating the short-lived gamma-frequency oscillations generated within neuronal circuits - coupled according to the diffusion weighted tractography between brain areas. Varying the global coupling strength and conduction speed, we identify a critical regime where spatially and spectrally resolved metastable oscillatory modes (MOMs) emerge at sub-gamma frequencies, approximating the MEG power spectra from 89 healthy individuals at rest. Further, we demonstrate that the frequency, duration, and scale of MOMs - as well as the frequency-specific envelope functional connectivity - can be controlled by global parameters, while the connectome structure remains unchanged. Grounded in the physics of delay-coupled oscillators, these numerical analyses demonstrate how interactions between locally generated fast oscillations in the connectome spacetime structure can lead to the emergence of collective brain rhythms organized in space and time.
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Affiliation(s)
- Joana Cabral
- Life and Health Sciences Research Institute (ICVS), School of Medicine, University of Minho, Braga, Portugal
- Centre for Eudaimonia and Human Flourishing, Linacre College, University of Oxford, Oxford, UK
- Center for Music in the Brain, Department of Clinical Medicine, Aarhus University, Aarhus, Denmark
- ICVS/3B’s - Portuguese Government Associate Laboratory, Braga/Guimarães, Portugal
| | - Francesca Castaldo
- Wellcome Centre for Human Neuroimaging, University College London, Queen Square Institute of Neurology, London, UK
| | - Jakub Vohryzek
- Centre for Eudaimonia and Human Flourishing, Linacre College, University of Oxford, Oxford, UK
- Center for Brain and Cognition, Computational Neuroscience Group, Universitat Pompeu Fabra, Barcelona, Spain
| | - Vladimir Litvak
- Wellcome Centre for Human Neuroimaging, University College London, Queen Square Institute of Neurology, London, UK
| | - Christian Bick
- Department of Mathematics, Vrije Universiteit Amsterdam, Amsterdam, The Netherlands
- Amsterdam Neuroscience – Systems & Network Neuroscience, Amsterdam, The Netherlands
- Mathematical Institute, University of Oxford, Oxford, UK
- Department of Mathematics, University of Exeter, Exeter, UK
| | | | - Karl Friston
- Wellcome Centre for Human Neuroimaging, University College London, Queen Square Institute of Neurology, London, UK
| | - Morten L. Kringelbach
- Life and Health Sciences Research Institute (ICVS), School of Medicine, University of Minho, Braga, Portugal
- Centre for Eudaimonia and Human Flourishing, Linacre College, University of Oxford, Oxford, UK
- Center for Music in the Brain, Department of Clinical Medicine, Aarhus University, Aarhus, Denmark
- Department of Psychiatry, University of Oxford, Oxford, UK
| | - Gustavo Deco
- Center for Brain and Cognition, Computational Neuroscience Group, Universitat Pompeu Fabra, Barcelona, Spain
- Institució Catalana de la Recerca i Estudis Avançats (ICREA), Barcelona, Spain
- Department of Neuropsychology, Max Planck Institute for Human Cognitive and Brain Sciences, Leipzig, Germany
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Brezetsky S, Jaros P, Levchenko R, Kapitaniak T, Maistrenko Y. Chimera complexity. Phys Rev E 2021; 103:L050204. [PMID: 34134258 DOI: 10.1103/physreve.103.l050204] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/19/2020] [Accepted: 04/23/2021] [Indexed: 06/12/2023]
Abstract
We show an amazing complexity of the chimeras in small networks of coupled phase oscillators with inertia. The network behavior is characterized by heteroclinic switching between multiple saddle chimera states and riddling basins of attractions, causing an extreme sensitivity to initial conditions and parameters. Additional uncertainty is induced by the presumable coexistence of stable phase-locked states or other stable chimeras as the switching trajectories can eventually tend to them. The system dynamics becomes hardly predictable, while its complexity represents a challenge in the network sciences.
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Affiliation(s)
- Serhiy Brezetsky
- Division of Dynamics, Lodz University of Technology, Stefanowskiego 1/15, 90-924 Lodz, Poland
| | - Patrycja Jaros
- Division of Dynamics, Lodz University of Technology, Stefanowskiego 1/15, 90-924 Lodz, Poland
| | - Roman Levchenko
- Faculty of Radiophysics, Electronics and Computer Systems, Taras Shevchenko National University of Kyiv, Volodymyrska St. 60, 01030 Kyiv, Ukraine
- Forschungszentrum Jülich, 52428 Jülich, Germany
| | - Tomasz Kapitaniak
- Division of Dynamics, Lodz University of Technology, Stefanowskiego 1/15, 90-924 Lodz, Poland
| | - Yuri Maistrenko
- Division of Dynamics, Lodz University of Technology, Stefanowskiego 1/15, 90-924 Lodz, Poland
- Forschungszentrum Jülich, 52428 Jülich, Germany
- Institute of Mathematics and Centre for Medical and Biotechnical Research, NAS of Ukraine, Tereshchenkivska St. 3, 01030 Kyiv, Ukraine
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Kuehn C, Bick C. A universal route to explosive phenomena. SCIENCE ADVANCES 2021; 7:7/16/eabe3824. [PMID: 33863722 PMCID: PMC8051866 DOI: 10.1126/sciadv.abe3824] [Citation(s) in RCA: 17] [Impact Index Per Article: 5.7] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 08/18/2020] [Accepted: 02/25/2021] [Indexed: 06/12/2023]
Abstract
Critical transitions are observed in many complex systems. This includes the onset of synchronization in a network of coupled oscillators or the emergence of an epidemic state within a population. "Explosive" first-order transitions have caught particular attention in a variety of systems when classical models are generalized by incorporating additional effects. Here, we give a mathematical argument that the emergence of these first-order transitions is not surprising but rather a universally expected effect: Varying a classical model along a generic two-parameter family must lead to a change of the criticality. To illustrate our framework, we give three explicit examples of the effect in distinct physical systems: a model of adaptive epidemic dynamics, for a generalization of the Kuramoto model, and for a percolation transition.
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Affiliation(s)
- Christian Kuehn
- Faculty of Mathematics, Technical University of Munich, Garching, Germany
- Complexity Science Hub Vienna, Vienna, Austria
| | - Christian Bick
- Department of Mathematics, University of Exeter, Exeter, UK.
- Department of Mathematics, Vrije Universiteit Amsterdam, Amsterdam, Netherlands
- Institute for Advanced Study, Technical University of Munich, Garching, Germany
- Mathematical Institute, University of Oxford, Oxford, UK
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Bick C, Goodfellow M, Laing CR, Martens EA. Understanding the dynamics of biological and neural oscillator networks through exact mean-field reductions: a review. JOURNAL OF MATHEMATICAL NEUROSCIENCE 2020; 10:9. [PMID: 32462281 PMCID: PMC7253574 DOI: 10.1186/s13408-020-00086-9] [Citation(s) in RCA: 91] [Impact Index Per Article: 22.8] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/17/2019] [Accepted: 05/07/2020] [Indexed: 05/03/2023]
Abstract
Many biological and neural systems can be seen as networks of interacting periodic processes. Importantly, their functionality, i.e., whether these networks can perform their function or not, depends on the emerging collective dynamics of the network. Synchrony of oscillations is one of the most prominent examples of such collective behavior and has been associated both with function and dysfunction. Understanding how network structure and interactions, as well as the microscopic properties of individual units, shape the emerging collective dynamics is critical to find factors that lead to malfunction. However, many biological systems such as the brain consist of a large number of dynamical units. Hence, their analysis has either relied on simplified heuristic models on a coarse scale, or the analysis comes at a huge computational cost. Here we review recently introduced approaches, known as the Ott-Antonsen and Watanabe-Strogatz reductions, allowing one to simplify the analysis by bridging small and large scales. Thus, reduced model equations are obtained that exactly describe the collective dynamics for each subpopulation in the oscillator network via few collective variables only. The resulting equations are next-generation models: Rather than being heuristic, they exactly link microscopic and macroscopic descriptions and therefore accurately capture microscopic properties of the underlying system. At the same time, they are sufficiently simple to analyze without great computational effort. In the last decade, these reduction methods have become instrumental in understanding how network structure and interactions shape the collective dynamics and the emergence of synchrony. We review this progress based on concrete examples and outline possible limitations. Finally, we discuss how linking the reduced models with experimental data can guide the way towards the development of new treatment approaches, for example, for neurological disease.
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Affiliation(s)
- Christian Bick
- Centre for Systems, Dynamics, and Control, University of Exeter, Exeter, UK.
- Department of Mathematics, University of Exeter, Exeter, UK.
- EPSRC Centre for Predictive Modelling in Healthcare, University of Exeter, Exeter, UK.
- Mathematical Institute, University of Oxford, Oxford, UK.
- Institute for Advanced Study, Technische Universität München, Garching, Germany.
| | - Marc Goodfellow
- Department of Mathematics, University of Exeter, Exeter, UK
- EPSRC Centre for Predictive Modelling in Healthcare, University of Exeter, Exeter, UK
- Living Systems Institute, University of Exeter, Exeter, UK
- Wellcome Trust Centre for Biomedical Modelling and Analysis, University of Exeter, Exeter, UK
| | - Carlo R Laing
- School of Natural and Computational Sciences, Massey University, Auckland, New Zealand
| | - Erik A Martens
- Department of Applied Mathematics and Computer Science, Technical University of Denmark, Kgs. Lyngby, Denmark.
- Department of Biomedical Science, University of Copenhagen, Copenhagen N, Denmark.
- Centre for Translational Neuroscience, University of Copenhagen, Copenhagen N, Denmark.
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Omel'chenko OE. Nonstationary coherence-incoherence patterns in nonlocally coupled heterogeneous phase oscillators. CHAOS (WOODBURY, N.Y.) 2020; 30:043103. [PMID: 32357679 DOI: 10.1063/1.5145259] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/16/2020] [Accepted: 03/16/2020] [Indexed: 06/11/2023]
Abstract
We consider a large ring of nonlocally coupled phase oscillators and show that apart from stationary chimera states, this system also supports nonstationary coherence-incoherence patterns (CIPs). For identical oscillators, these CIPs behave as breathing chimera states and are found in a relatively small parameter region only. It turns out that the stability region of these states enlarges dramatically if a certain amount of spatially uniform heterogeneity (e.g., Lorentzian distribution of natural frequencies) is introduced in the system. In this case, nonstationary CIPs can be studied as stable quasiperiodic solutions of a corresponding mean-field equation, formally describing the infinite system limit. Carrying out direct numerical simulations of the mean-field equation, we find different types of nonstationary CIPs with pulsing and/or alternating chimera-like behavior. Moreover, we reveal a complex bifurcation scenario underlying the transformation of these CIPs into each other. These theoretical predictions are confirmed by numerical simulations of the original coupled oscillator system.
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Affiliation(s)
- Oleh E Omel'chenko
- Institute of Physics and Astronomy, University of Potsdam, Karl-Liebknecht-Str. 24/25, 14476 Potsdam, Germany
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Ashwin P, Bick C, Poignard C. State-dependent effective interactions in oscillator networks through coupling functions with dead zones. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2019; 377:20190042. [PMID: 31656136 PMCID: PMC6833998 DOI: 10.1098/rsta.2019.0042] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Accepted: 09/02/2019] [Indexed: 06/10/2023]
Abstract
The dynamics of networks of interacting dynamical systems depend on the nature of the coupling between individual units. We explore networks of oscillatory units with coupling functions that have 'dead zones', that is the coupling functions are zero on sets with interior. For such networks, it is convenient to look at the effective interactions between units rather than the (fixed) structural connectivity to understand the network dynamics. For example, oscillators may effectively decouple in particular phase configurations. Along trajectories, the effective interactions are not necessarily static, but the effective coupling may evolve in time. Here, we formalize the concepts of dead zones and effective interactions. We elucidate how the coupling function shapes the possible effective interaction schemes and how they evolve in time. This article is part of the theme issue 'Coupling functions: dynamical interaction mechanisms in the physical, biological and social sciences'.
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León I, Pazó D. Phase reduction beyond the first order: The case of the mean-field complex Ginzburg-Landau equation. Phys Rev E 2019; 100:012211. [PMID: 31499758 DOI: 10.1103/physreve.100.012211] [Citation(s) in RCA: 26] [Impact Index Per Article: 5.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/17/2019] [Indexed: 05/01/2023]
Abstract
Phase reduction is a powerful technique that makes possible to describe the dynamics of a weakly perturbed limit-cycle oscillator in terms of its phase. For ensembles of oscillators, a classical example of phase reduction is the derivation of the Kuramoto model from the mean-field complex Ginzburg-Landau equation (MF-CGLE). Still, the Kuramoto model is a first-order phase approximation that displays either full synchronization or incoherence, but none of the nontrivial dynamics of the MF-CGLE. This fact calls for an expansion beyond the first order in the coupling constant. We develop an isochron-based scheme to obtain the second-order phase approximation, which reproduces the weak-coupling dynamics of the MF-CGLE. The practicality of our method is evidenced by extending the calculation up to third order. Each new term of the power-series expansion contributes with additional higher-order multibody (i.e., nonpairwise) interactions. This points to intricate multibody phase interactions as the source of pure collective chaos in the MF-CGLE at moderate coupling.
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Affiliation(s)
- Iván León
- Instituto de Física de Cantabria (IFCA), CSIC-Universidad de Cantabria, 39005 Santander, Spain
| | - Diego Pazó
- Instituto de Física de Cantabria (IFCA), CSIC-Universidad de Cantabria, 39005 Santander, Spain
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Kasatkin DV, Klinshov VV, Nekorkin VI. Itinerant chimeras in an adaptive network of pulse-coupled oscillators. Phys Rev E 2019; 99:022203. [PMID: 30934254 DOI: 10.1103/physreve.99.022203] [Citation(s) in RCA: 12] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/13/2018] [Indexed: 11/07/2022]
Abstract
In a network of pulse-coupled oscillators with adaptive coupling, we discover a dynamical regime which we call an "itinerant chimera." Similarly as in classical chimera states, the network splits into two domains, the coherent and the incoherent. The drastic difference is that the composition of the domains is volatile, i.e., the oscillators demonstrate spontaneous switching between the domains. This process can be seen as traveling of the oscillators from one domain to another or as traveling of the chimera core across the network. We explore the basic features of the itinerant chimeras, such as the mean and the variance of the core size, and the oscillators lifetime within the core. We also study the scaling behavior of the system and show that the observed regime is not a finite-size effect but a key feature of the collective dynamics which persists even in large networks.
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Affiliation(s)
- Dmitry V Kasatkin
- Institute of Applied Physics of the Russian Academy of Sciences, 46 Ul'yanov Street, 603950, Nizhny Novgorod, Russia
| | - Vladimir V Klinshov
- Institute of Applied Physics of the Russian Academy of Sciences, 46 Ul'yanov Street, 603950, Nizhny Novgorod, Russia
| | - Vladimir I Nekorkin
- Institute of Applied Physics of the Russian Academy of Sciences, 46 Ul'yanov Street, 603950, Nizhny Novgorod, Russia
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